Randomly stopped sums: models and psychological applications

Abstract

This paper describes an approach to modeling the sums of a continuous random variable over a number of measurement occasions when the number of occasions also is a random variable. A typical example is summing the amounts of time spent attending to pieces of information in an information search task leading to a decision to obtain the total time taken to decide. Although there is a large literature on randomly stopped sums in financial statistics, it is largely absent from psychology. The paper begins with the standard modeling approaches used in financial statistics, and then extends them in two ways. First, the randomly stopped sums are modeled as “life distributions” such as the gamma or log-normal distribution. A simulation study investigates Type I error rate accuracy and power for gamma and log-normal versions of this model. Second, a Bayesian hierarchical approach is used for constructing an appropriate general linear model of the sums. Model diagnostics are discussed, and three illustrations are presented from real datasets.

A Stochastic Condensation Mechanism for Inducing Underdispersion in Count Models

“…The change of variable t = r − m yields on using r = m + t = m − |t| (since t < 0) (c): The proposition (P ) can be proved by induction on j as follows. For j = 0, we have H y (y) = f c (y|µ, θ, m), and this is consistent with (P ) by Equation (10). Next, notice from the definition of H y (j) that H y (k + 1) = f c (y − k − 1|µ, θ, m) + H y (k).…”

“…Poisson-log normal [8] and Poisson-inverse Gaussian distributions are obtained by the same approach [9]. Randomly-stopped sum [10,11] and compounding are less known mechanisms which also leads to overdispersion [3].…”

Regression for copula-linked compound distributions with applications in modeling aggregate insurance claims

“…In operational risk, the compound distribution for aggregate losses is the foundation for the determination of the operational risk capital required by the Basel capital framework for banks (Panjer (2006) and Shevchenko (2010)). In psychology, Smithson and Shou (2014) pointed out the applications of this type of model in different areas of psychology such as perception and decision making, where a psychological process is thought to be serially summed from observable component process outputs.…”